{{Short description|Class of convex shapes}} {{Use mdy dates|cs1-dates=ly|date=December 2024}} {{Use list-defined references|date=December 2024}} {{CS1 config|mode=cs2}} In convex geometry, a '''zonoid''' is a type of centrally symmetric convex body.
==Definitions== The zonoids have several definitions, equivalent up to translations of the resulting shapes:{{r|b69}}
* A zonoid is a shape that can be approximated arbitrarily closely (in Hausdorff distance) by a zonotope, a convex polytope formed from the Minkowski sum of finitely many line segments. In particular, every zonotope is a zonoid.{{r|b69}} Approximating a zonoid to within Hausdorff distance <math>\varepsilon</math> requires a number of segments that (for fixed <math>\varepsilon</math>) is near-linear in the dimension, or linear with some additional assumptions on the zonoid.{{r|blm}} * A zonoid is the range of an atom-free vector-valued sigma-additive set function. Here, a function from a family of sets to vectors is sigma-additive when the family is closed under countable disjoint unions, and when the value of the function on a union of sets equals the sum of its values on the sets. It is atom-free when every set whose function value is nonzero has a proper subset whose value remains nonzero. For this definition the resulting shapes contain the origin, but they may be translated arbitrarily as long as they contain the origin.{{r|b69}} The statement that the shapes described in this way are closed and convex is known as Lyapunov's theorem. * A zonoid is the convex hull of the range of a vector-valued sigma-additive set function. For this definition, being atom-free is not required.{{r|b69}} * A zonoid is the polar body of a central section of the unit ball of <math>L^1([0,1])</math>, the space of Lebesgue integrable functions on the unit interval. Here, a central section is the intersection of this ball with a finite-dimensional subspace of <math>L^1([0,1])</math>. This definition produces zonoids whose center of symmetry is at the origin.{{r|b69}} * A zonoid is a convex set whose polar body is a projection body.{{r|b69}}
==Examples== Every two-dimensional centrally-symmetric convex shape is a zonoid.{{r|b71}} In higher dimensions, the Euclidean unit ball is a zonoid.{{r|b69}} A polytope is a zonoid if and only if it is a zonotope.{{r|blm}} Thus, for instance, the regular octahedron is an example of a centrally symmetric convex shape that is not a zonoid.{{r|b69}}
The solid of revolution of the positive part of a sine curve is a zonoid, obtained as a limit of zonohedra whose generating segments are symmetric to each other with respect to rotations around a common axis.{{r|cc}} The bicones provide examples of centrally symmetric solids of revolution that are not zonoids.{{r|b69}}
==Properties== Zonoids are closed under affine transformations,{{r|blm}} under parallel projection,{{r|rz}} and under finite Minkowski sums. Every zonoid that is not a line segment can be decomposed as a Minkowski sum of other zonoids that do not have the same shape as the given zonoid. (This means that they are not translates of homothetes of the given zonoid.){{r|b69}}
The zonotopes can be characterized as polytopes having centrally-symmetric pairs of opposite faces, and the ''zonoid problem'' is the problem of finding an analogous characterization of zonoids. Ethan Bolker credits the formulation of this problem to a 1916 publication of Wilhelm Blaschke.{{r|b71}}
==References== <references>
<ref name=b69>{{citation | last = Bolker | first = Ethan D. | doi = 10.2307/1995073 | journal = Transactions of the American Mathematical Society | mr = 256265 | pages = 323–345 | title = A class of convex bodies | volume = 145 | year = 1969| jstor = 1995073 }}</ref>
<ref name=b71>{{citation | last = Bolker | first = E. D. | department = Research Problems | doi = 10.2307/2317764 | issue = 5 | journal = The American Mathematical Monthly | jstor = 2317764 | mr = 1536334 | pages = 529–531 | title = The zonoid problem | volume = 78 | year = 1971}}</ref>
<ref name=blm>{{citation | last1 = Bourgain | first1 = J. | author1-link = Jean Bourgain | last2 = Lindenstrauss | first2 = J. | author2-link = Joram Lindenstrauss | last3 = Milman | first3 = V. | author3-link = Vitali Milman | doi = 10.1007/BF02392835 | issue = 1–2 | journal = Acta Mathematica | mr = 981200 | pages = 73–141 | title = Approximation of zonoids by zonotopes | volume = 162 | year = 1989}}</ref>
<ref name=cc>{{citation | last1 = Chilton | first1 = B. L. | last2 = Coxeter | first2 = H. S. M. | author2-link = Harold Scott MacDonald Coxeter | doi = 10.2307/2313051 | journal = The American Mathematical Monthly | jstor = 2313051 | mr = 157282 | pages = 946–951 | title = Polar zonohedra | volume = 70 | year = 1963| issue = 9 }}</ref>
<ref name=rz>{{citation | last1 = Ryabogin | first1 = Dmitry | last2 = Zvavitch | first2 = Artem | contribution = Analytic methods in convex geometry | contribution-url = https://www.impan.pl/dzialalnosc/studia-doktoranckie/studia/special-lectures-for-phd-students/rz2011.pdf | isbn = 978-83-86806-24-9 | mr = 3329057 | pages = 87–183 | publisher = Polish Acad. Sci. Inst. Math., Warsaw | series = IMPAN Lect. Notes | title = Analytical and probabilistic methods in the geometry of convex bodies | volume = 2 | year = 2014 | access-date = 2024-12-08 | archive-date = 2024-12-17 | archive-url = https://web.archive.org/web/20241217105101/https://www.impan.pl/dzialalnosc/studia-doktoranckie/studia/special-lectures-for-phd-students/rz2011.pdf | url-status = dead }}; see in particular section 4, "Zonoids and zonotopes"</ref>
</references>
==Further reading== *{{citation | last1 = Goodey | first1 = Paul | last2 = Weil | first2 = Wolfgang | author2-link = Wolfgang Weil (mathematician) | editor1-last = Gruber | editor1-first = Peter M. | editor2-last = Wills | editor2-first = Jörg M. | contribution = Zonoids and generalisations | doi = 10.1016/b978-0-444-89597-4.50020-2 | isbn = 9780444895974 | pages = 1297–1326 | publisher = Elsevier | title = Handbook of Convex Geometry | volume = B | year = 1993}} *{{citation | last1 = Schneider | first1 = Rolf | last2 = Weil | first2 = Wolfgang | editor1-last = Gruber | editor1-first = Peter M. | editor2-last = Wills | editor2-first = Jörg M. | contribution = Zonoids and related topics | doi = 10.1007/978-3-0348-5858-8_13 | isbn = 9783034858588 | location = Basel | pages = 296–317 | publisher = Birkhäuser | title = Convexity and Its Applications | year = 1983}}
Category:Convex geometry